Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]
What fractions can you find between the square roots of 65 and 67?
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
This printable worksheet may be useful: Tet Trouble.
In silence, write three lengths on the board (for example 3 units, 6 units, 7units) and accurately draw a triangle with sides of corresponding lengths. You could use a dynamic geometry package to do this.
Do it again with three more lengths.
And again but instead of drawing the triangle put a question mark. After some thinking time, encourage a member of the group to come up and draw the triangle.
Finally, list three lengths that will not work followed by a question mark and after time has been taken to realise the impossibility, discuss why this is the case as a group.
Now pose the problem.
Working in small groups the challenge will be to employ systematic approaches as well as applying the triangle inequality.
Take opportunities to pull together different ideas for recording, including the use of nets and working systematically.